The Hands On The Clock

Overview and Objective

In this lesson, students volition utilize the geared clocks on Polypad to explore the angle patterns on an analog clock. Specifically, they will find the number of times in a day that a clock's minute and hour hands overlap to form a straight line.

Warm-Upwards

Invite students to open a blank sail and use the geared clock to adjust so that the clock's minute and hour easily course a straight angle.

Although students tin come upwardly with different answers, the obvious answer is six o'clock. Clarify with the students that for a.thousand /p.thou fashion, that are ii different times a day for 6 a.m and half-dozen p.grand. In 24 hours fashion, that is 6:00 and eighteen:00.

Therefore in a 24-60 minutes twenty-four hours, we already found two examples of times that the infinitesimal and hour paw create a $180^o$ bending.

You may suggest students employ the toggle seconds selection at the bottom center toolbar later selecting a clock on the canvas to hibernate and show the second hand.

If the students plant the other directly angles, let them share their answers by recording those times on a board. If not, inquire them to movement the infinitesimal paw slowly to observe the other times that create $180^o$ angle. Hash out with students that information technology is non piece of cake to tell the time precisely that creates the straight angle other than 6 a.thousand and p.m.

Principal Activity

Using the tabular array tool, take students tape all possible times with $180^o$ bending (between 6 a.m and six p.thousand). Students may also use the protractor tool if they need information technology. Even so, remind them they do not demand to be precise here since they are actually looking at the number of times that hands form a straight bending rather than the actual time. Approximation for each time is enough.

Here are some possible answers.

Students could talk near patterns they notice, such as

In 12 hours, in that location are 11 times they tin can observe a straight angle.
Later on 6 a.m, almost every 65 minutes, they run across a directly angle.

Hither they can employ their observations to create an equation to discover the exact times.

11 equal parts in 12 hours mean 11 equal parts in 720 minutes. It ways, in every

${720 \over xi}$ = $65$ ${5 \over eleven}$ minutes (which is very shut to their interpretation), they can observe a straight angle on an analog clock. While intuition might tell us that there should be 24, nosotros proved that there are 22 times a solar day a straight angle can exist observed the clock.

They tin too find the verbal times by adding 65 ${5 \over 11}$ minutes to 6:00. To brand the calculations easier, we tin can express each interval as 65 and ${5 \over 11}$ minutes which is

= 65.454545454.. minutes = 1h 5min 27.272727..seconds

Another approach to finding times tin can be using the unit rates.

Analyze with students that the minute paw coves 360 degrees (makes a full turn) in threescore minutes. Discuss the speed of infinitesimal paw in degrees per minute. The hour mitt covers 360 degrees in 12 hours (= 720 minutes), and inquire well-nigh the speed of hour manus in caste per hour and minute.

The speed of the infinitesimal hand is 6 degrees per infinitesimal, whereas the speed of the hour hand is 30 degrees per hour which ways $1/two$ degrees per minute.

We know that at 6:00, both are at the same position simply will now movement at unlike speeds.

The relative speed between the minute and 60 minutes hand = $vi - i/2 = eleven/2$ caste per infinitesimal.

They will create some other straight angle when the relative speed between them covers 360 degrees.

Therefore in every ${360 \over eleven/two}$ which means ${720 \over 11}$ minutes, they will create some other 180-degree angle.

In the second part of the lesson, after discussing the relative speeds of the hour of minute hands, you tin can ask about how many times a day the hour and minute manus overlaps?

Once more start with the about obvious one, which is 12 o'clock. Permit them apply different clocks to show each time with overlapping hands.

Closure

Invite a unlike student to share each time with the overlapping easily. To close the lesson, ask students to try to find the verbal times with the overlapping hands. They may write the times over a 12-time period then convert to a.grand /p.m versions.